View source for Regression spectrum

The spectrum of a stochastic process occurring in the regression scheme for a stationary time series. Thus, let a [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806501.png" /> which is observable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806502.png" /> be represented in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806504.png" /> is a [[Stationary stochastic process|stationary stochastic process]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806505.png" />, and let the mean value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806506.png" /> be expressed in the form of a linear [[Regression|regression]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806509.png" />, are known regression vectors and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065010.png" /> are unknown regression coefficients (cf. [[Regression coefficient|Regression coefficient]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065011.png" /> be the spectral distribution function of the regression vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065012.png" /> (cf. [[Spectral analysis of a stationary stochastic process|Spectral analysis of a stationary stochastic process]]). The regression spectrum for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065013.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065015.png" /> for any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065016.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065018.png" />.

The regression spectrum plays an important role in problems of estimating the regression coefficients in the scheme (1)–(2). For example, the elements of a regression spectrum can be used to express a necessary and sufficient condition for the asymptotic efficiency of an estimator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065019.png" /> by the method of least squares.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  U. Grenander,   M. Rosenblatt,   "Statistical analysis of stationary time series" , Wiley  (1957)</TD></TR></table>